Inhomogeneous diophantine approximations and distribution of fractional parts (Q690576)

Let \(\alpha\) be an irrational number with continued fraction expansion \(\alpha=[0;q_1,q_2,\dots]\), \(I\) be an interval in \([0,1)\) with the length \(|I|\), \(\{n\alpha\}\) be a fractional part of \(n\alpha\), \(n=1,2,\dots\), and denote the local discrepancy function \(r(\alpha,a,n,I)=|\#\{1\leq k\leq n;\{k\alpha + a\}\in I\}-n|I||\). Let \(|I|=\beta\). In the present paper the author obtain a bound for \(r(\alpha,a,n,I)\), considering the sequence of best \(\alpha\)-approximations to \(\beta\) as follows: \[ A_{\alpha\beta}(0)=\begin{cases} 0&\text{if }\beta\leq\frac{1}{2}\\ 1&\text{if }\beta>\frac{1}{2}\end{cases}; \] \[ A_{\alpha\beta}(m+1)=\min\{k\in\mathbb N;\|k\alpha-\beta\|<\|A_{\alpha\beta}(m)\alpha-\beta\|\}; \] \[ a_{\alpha\beta}(n)=\min\{m;A_{\alpha\beta}(m)>n\}; \] \[ \tilde a(n)=A_{\alpha\beta}(a_{\alpha\beta}(n)); \] \[ \psi(n)=\bigg[\log_{\frac{1+\sqrt{5}+1}{2}}\bigg(\sqrt{5}n+\sqrt{5}+\frac{1}{2}\bigg)\bigg]+1. \] The main result of the present paper is \[ r(\alpha,a,n,I)\leq\frac{1}{2}\bigg(\max_{1\leq j\leq\psi(\tilde a(n))+2}q_j+15\bigg)(a_{\alpha\beta}(n)+1). \] In the proof the author uses generalized Fibonacci tilings as in his previous papers [in: Analytic and probabilistic methods in number theory. Proceedings of the 4th international conference in honour of J. Kubilius, Palanga, Lithuania, September 25--29, 2006. Vilnius: TEV. 190--203 (2007; Zbl 1165.11062)] and [Chebyshevskiĭ Sb. 7, No. 3(19), 110--128 (2006; Zbl 1241.11091)].